User gfr - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T22:48:47Z http://mathoverflow.net/feeds/user/29850 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/130740/circle-bundles-over-cp1-and-self-intersection-number-of-cp1-embeddings Circle bundles over $CP^1$ and self-intersection number of $CP^1$ embeddings GFR 2013-05-15T17:00:49Z 2013-05-16T09:35:56Z <p>If $X$ is a compact oriented surface in a 4-dimensional oriented manifold $M$, then the self-intersection number $X^2$ of $X$ is given by the integral over $X$ of the Euler class of the normal bundle. In the case of $CP^1$ embedded in $CP^2$, the normal bundle is isomorphic to the Hopf bundle, therefore $X^2$ can be obtained calculating the first Chern number of the Hopf fibration (or equivalently the Euler number of its realization).</p> <p>It is possible to have circle bundles on $CP^1$ with higher Chern number by taking the quotient of the total space of the Hopf fibration by the action generated by $(z^1,z^2)\mapsto(z^1 \exp(i 2 \pi/k), z^2 \exp(i 2\pi/k ))$.</p> <p>Are these bundles the normal bundles of some embedding of $CP^1$ in a 4-dimensional manifold? If yes, is it possible to describe the embedding explicitly? Is there a deeper relation between the Hopf bundle and the normal bundle of $CP^1$ embedded in $CP^2$ or do they just happen to be the same?</p> http://mathoverflow.net/questions/125266/sign-convention-in-generalised-gauss-bonnet Sign convention in generalised Gauss-Bonnet GFR 2013-03-22T11:35:42Z 2013-03-22T11:58:58Z <p>Apparently I cannot get the right sign in deriving classical Gauss-Bonnet from generalised one!</p> <p>According to many references (e.g. Madsen and Tornehave, Nakahara, Milnor and Stasheff if you don't use their particular convention about orientation) generalised Gauss-Bonnet is</p> <p>$\int_M Pf(-R/2\pi)=\chi(M)$</p> <p>where $R$ is the curvature 2-form and the Pfaffian of a 2lx2l antisymmetric matrix is $Pf(A)=\frac{1}{2^ll!}\epsilon ^{i_1 \ldots i_{2l}}A_{i_1 i_2} \ldots A_{i_{2l-1}i_{2l}} \$ i.e. the sign convention is such that for the matrix A={{0,a},{-a,0}} Pf(A)=a.</p> <p>For l=1, M is a surface and $R^a_{\phantom{a}b}=\frac{1}{2}R^a_{\phantom{a}bcd}e^c\wedge e^d =K (g_{ac}g_{bd}-g_{ad}g_{bc})$ where $K$ is the Gaussian curvature, $g_{ab}$ the metric tensor and ${e^a}$ an orthonormal basis. For the Pfaffian I get $Pf(-R/2 \pi)= -\frac{1}{4\pi} 2 \epsilon_{12} R^1_2= -\frac{1}{2\pi} R^1_{\phantom{1}2}=-\frac{1}{2\pi}R^1_{\phantom{1}212}e^1\wedge e^2$</p> <p>$e^1\wedge e^2$ is the volume form and $R^1_{\phantom{1}212}=K$ therefore</p> <p>$\int_M Pf(-R/2\pi)=\int_M-K/2\pi=\chi$</p> <p>which obviously has the wrong sign! I have carefully checked the definitions in the books and I think I am using the same as they use - so please be very explicit in pointing aout where the mistake is! Thanks</p> http://mathoverflow.net/questions/124527/understanding-four-manifolds-more-details-inside Understanding four manifolds (more details inside) GFR 2013-03-14T14:57:38Z 2013-03-14T18:35:48Z <p>I need to understand some of the theory of smooth four manifolds. Eventually I might be interested in learning about Donaldson theory. For the moment I am mainly interested in question such as: if you have a compact, connected, simply-connected four manifold, in how many different ways can I embed a smooth, compact 2-surface in it? A prototype question would be: how many different smooth embeddings of S^2 in the connected sum of CP^2 can I have? I would also like to learn how to calculate self-intersection numbers of 2-surfaces embedded in 4-manifolds.</p> <p>I have some knowledge of differential geometry and general topology and, at a much lower level, of algebraic topology but my background is in theoretical physics. To give an idea, I feel at home with books like Nakahara "Geometry topology and physics" or "Gravitation" by Misner, Thorne and Wheeler, I like and can follow the more mathematical Naber "Topology, geometry and gauge fields", I find more challenging a book like Bott, Tu "Differential forms in algebraic topology", but I have not worked through it yet (I am more familiar with de Rham theory than anything else in algebraic topology) but e.g. Hatcher "Algebraic Topology" is really though for me.</p> <p>Given my background, what route would you recommend to learn enough material to allow me to calculate things like those I have pointed out above? I do not want to do research in this area, but be familiar enough with it so that I can use it-see the questions I have raised above for an example of what I mean.</p> http://mathoverflow.net/questions/116074/natural-connection-on-u1-principal-bundles-over-s2-with-chern-number1 Natural connection on U(1) principal bundles over S^2 with Chern number>1 GFR 2012-12-11T12:16:20Z 2012-12-11T12:32:17Z <p>S^3 can be seen as a U(1)-bundle over S^2 (Hopf fibration). It has first Chern number 1 or -1. Denoting with z_1,z_2 coordinates on C^2 and restricting to S^3 the natural connection form is \omega_1=\bar z_1 dz_1+\bar z_2 dz_2</p> <p>Consider now the U(1) bundles over S^2 with Chern number n>1. They have total space S^3/Z_n where the Z_n action on C^2 is generated by (z_1,z_2)->(z_1\exp(i2\pi/n),z_2\exp(i2\pi/n)). I know that the appropriate connection form which generalises the one of the Hopf bundle is \omega_n=n(\bar z_1 dz_1+\bar z_2 dz_2) I do not understand where the factor of n comes from. In other terms how do I see, from the defining properties of a connection form (or otherwise), that a factor of n is needed and/or that any other factor would give a one-form which is not a connection form?</p> http://mathoverflow.net/questions/125266/sign-convention-in-generalised-gauss-bonnet Comment by GFR GFR 2013-03-22T15:34:20Z 2013-03-22T15:34:20Z It turned out that was the problem indeed: my convention for the definition of curvature 2-form where different from those used in the books, resulting in a minus sign difference. http://mathoverflow.net/questions/125266/sign-convention-in-generalised-gauss-bonnet/125267#125267 Comment by GFR GFR 2013-03-22T15:27:15Z 2013-03-22T15:27:15Z I think I have found where the sign comes from: I was using conventions such that the connection form $\omega$ satisfies $de^i+\omega^i_{\phantom{i}j}\wedge e^j=0$, while Milnor-Stasheff has a minus sign, see pag.302. This basically correspond to replace $R^i_{\phantom{i}j}$ with $\R^j_{\phantom{i}i}=-R^i_{\phantom{i}j}$, so that an overall minus results if there is an odd number of $R^i_{\phantom{i}j}$s. http://mathoverflow.net/questions/125266/sign-convention-in-generalised-gauss-bonnet/125267#125267 Comment by GFR GFR 2013-03-22T15:21:02Z 2013-03-22T15:21:02Z That is true but hey also use an unconventional orientation: see page 304, note on signs. Basically if the manifold has dimension 2l and l is even their orientation is the same as the conventional one: $e^1\wedge \ldots \wedge e^{2l}$ but otherwise there is a minus sign. Quoting from the book: &quot;Readers who prefer to use the classical sign conventions as in [Spanier], [Warner] and [Bott-Chern] can forget about these signs, but should replace K with -K whenever it occurs in our characteristic class formulas&quot;. http://mathoverflow.net/questions/125266/sign-convention-in-generalised-gauss-bonnet/125267#125267 Comment by GFR GFR 2013-03-22T12:40:16Z 2013-03-22T12:40:16Z Thanks Liviu, I agree that with the conventions that you pointed out everything works out fine. I still have not accepted the answer to see if anyone can confirm that there is a mistake in the books I have mentioned or - more likely - point out what I missed out. In many sources the definition of the Pfaffian is given as $l!A\wedge \ldots \wedge A=Pf(A) vol$ which is equivalent to my definition above.